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For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4.

This is Mahattan's explanation.

Is there any other ways to solve this problem easily?

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

a. x = w b. x > w c. x/y is an integer d. w/z is an integer e. x/z is an integer

This is how you can work it out:

x = sum of y con. integers w = sum of z con. integers

Plug in values such that most of these are easily made true. Since z is in the denominator, I will try to put z = 1 to get w/z and x/z as integers without any complications.

If z = 1, y = 2 w = any one integer x = sum of 2 consecutive integers w/z and x/z will be integers so d and e can be true.

x = w is obviously possible. Say w = 9 and x = 4+5 x > w is also possible say if w = 8 while x = 4+5 Hence both a and b can be true.

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A.x = w B.x > w C.x/y is an integer D.w/z is an integer E.x/z is an integer

I will post OA later.

Basically C is the average of even number of consecutive terms ,

average of even number of consecutive terms is never an integer ( 1,2,3,4 or 4,5,6,7)

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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17 Oct 2012, 15:08

1

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x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

x = w x > w x/y is an integer w/z is an integer x/z is an integer

average1= x/y average2=w/z

y=2z,

implies that for every value of z y is even. ie. z=even say 4 y=2*4 =6 (even)

z=odd say 3

y=2*3 =6 (even)

the average of even numbers is not integer , so the answer is C.

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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25 Oct 2012, 04:19

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delta09 wrote:

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A. x = w B. x > w C. x/y is an integer D. w/z is an integer E. x/z is an integer

For these kind of questions, we need to parse through the options to find an option which can NEVER be true. There are three scenarios possible with any option: 1. We can easily find out that this is possible. Then, we can eliminate this option and move forward. 2. We can easily find out that the option is never true. Thus, we can stop here because this is the answer. 3. Nothing can be easily inferred. We need to plug in values and do some reasoning to figure out. These options can be skipped for the moment to come back later, if we don't find an answer among other options. We want to SAVE our time.

Here is my attempt at this: A. x = w - just looking at it, i can't say if this is possible or not. To find so will take time.I skip it to come back later B. x > w - Easily possible is x is the sum of first 2 integers and w is the sum of first 1 integer C. x/y is an integer - come back later D. w/z is an integer - easily possible if z=1, w=1 E. x/z is an integer[/quote] - easily possible if z=1, x=3

now, we are left with A and C options. I begin with option C.

x=sum of consecutive y (or 2z) integers = n + (n+1) + (n+2) +....+ (n+2z-1) = 2zn + (1+2+3+...+2z-1) = 2zn + 2z*(2z-1)/2 (using formula for sum of first n natural numbers) => x/y or x/2z = n + (2z-1)/2 we know n is an integer. The other component can never be an integer, since the numerator is always odd and denominator is 2. So, x/y cannot be an integer.

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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15 Aug 2013, 06:25

delta09 wrote:

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A. x = w B. x > w C. x/y is an integer D. w/z is an integer E. x/z is an integer

The fact is that y is always even and x is the sum of this y consecutive numbers.

like, x = 1+2+3+4 here y=4 again x= 6+7+8+9+10+11 and y=6 or x=1+2 and y=2 everywhere x/y = fraction
_________________

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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05 Mar 2014, 23:49

sum of Y consecutive values would be (2a1+(y-1)d)*y/2. since they are consecutive, d=1. so we have an equation (2a1+(y-1))*y/2 if we devide it by Y, we are left with (2a1+(y-1))/2=a1+(y-a)/2 where y-1 is and odd number and will not be devisible by 2, so it is not an integer

x is the sum of y consecutive integers. w is the sum of z [#permalink]

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17 May 2017, 18:26

kairoshan wrote:

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A. x = w B. x > w C. x/y is an integer D. w/z is an integer E. x/z is an integer

Say w = 1, x = 0 + 1. Then A works.

Say w = 1, x = 1 + 2. Then B works.

Say w = 1 + 2 + 3. Then D works.

Say w = 1 + 2 + 3, z=3, x = 1 + 2 + 3 + 4 + 5 + 6 = 21, x/z = 21 / 3 = 7. Then E works.

The trick here:

If x is the sum of 2z consecutive integers. Then x is the sum of an EVEN NUMBER of integers

But an even number of consecutive integers will have an odd sum

And you can't have Odd / Even = integer

It's actually a little more complicated than that because you could have, say, the sum of four consecutives.

But then you need the sum to divide by 4 Or if you have eight, you need it to divide by 8 and you'll keep coming up short a factor of 2 in your sum

But the basic idea is (1 + 2) doesn't divide by 2

(1 + 2 + 3 + 4) doesn't divide by 4 etc.

Answer : C _________________

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Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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11 Sep 2017, 09:06

sriharimurthy wrote:

For a set of 'n' consecutive integers, the sum of integers is : (a) always a multiple of 'n' when n is ODD (b) never a multiple of 'n' when n is EVEN (Note : Try it out!)

From y = 2z, we can conclude that x is the sum of consecutive integers for an EVEN number of terms since y will always be even.

Thus x/y can never be an integer.

Answer : C

thanks this is a brilliant explanation.

where could i get other number properties such as this?
_________________

For a set of 'n' consecutive integers, the sum of integers is : (a) always a multiple of 'n' when n is ODD (b) never a multiple of 'n' when n is EVEN (Note : Try it out!)

From y = 2z, we can conclude that x is the sum of consecutive integers for an EVEN number of terms since y will always be even.

Thus x/y can never be an integer.

Answer : C

thanks this is a brilliant explanation.

where could i get other number properties such as this?

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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11 Sep 2017, 20:07

h2polo wrote:

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

Case 1: x = w if y = [2,3] z = [5] 5=5, TRUE

Case 2: x > w if y = [2,3] z = [1] 5>1, TRUE

Case 3: x/y is an integer if y = [2,3,4] x = 9 9/3 is an integer, TRUE

Case 4: w/z is an integer if z = [2] w = 2 2/1 is an integer, TRUE

Case 5: x/z is an integer if x = 9, z = [2] 9/1 is an integer, TRUE

Am I doing something wrong here?

For case 3,

Y=2Z, so Y cannot equal 3, because then Z would equal 1.5 which isn't possible. Y must be even.

Re: x is the sum of y consecutive integers. w is the sum of z [#permalink]

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13 Sep 2017, 03:45

Ans Is C; Y = 2Z so y is an even number; We know the sum of N consecutive number is N(N+1)/2; So, sum of 2N(as y is even) consecutive number is 2N(2N+1)/2 = N(2N+1); We can't divide N(2N+1) by 2N, so X/Y is not possible.

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